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### OPTIMAL AND OPTIMUM TILINGS

The best tiling of a tree corresponds to an instruction sequence of least cost: the shortest sequence of instructions. Or if the instructions take different amounts of time to execute, the least-cost sequence has the lowest total time. Suppose we could give each kind of instruction a cost. Then we could define an optimum tiling as the one whose tiles sum to the lowest possible value. An optimal tiling is one where no two adjacent tiles can be combined into a single tile of lower cost. If there is some tree pattern that can be split into several tiles of lower combined cost, then we should remove that pattern from our catalog of tiles before we begin. Every optimum tiling is also optimal, but not vice versa. For example, suppose every instruction costs one unit, except for MOVEM, which costs m units. Then either Image 9.2a is optimum (if m > 1) or Image 9.2b is optimum (if m < 1) or both (if m = 1); but both trees are optimal.

Optimum tiling is based on an idealized cost model. In reality, instructions are not self-contained with individually attributable costs; nearby instructions interact in many ways, as discussed in .

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